# On well-posedness of the Muskat problem with surface tension

## TL;DR

This paper establishes the local well-posedness of the Muskat problem with surface tension for large initial data across all subcritical Sobolev spaces, covering various boundary conditions and dimensions, using a paradifferential approach.

## Contribution

It provides the first large-data well-posedness result for all subcritical Sobolev spaces in the Muskat problem with surface tension, including unbounded curvature and non-square integrable interfaces.

## Key findings

- Proves local well-posedness for large data in all subcritical Sobolev spaces.
- Handles interfaces with unbounded curvature and non-square integrable initial data.
- Reformulates the problem using Dirichlet-Neumann operators and paradifferential calculus.

## Abstract

We consider the Muskat problem with surface tension for one fluid or two fluids, with or without viscosity jump, with infinite depth or Lipschitz rigid boundaries, and in arbitrary dimension $d$ of the interface. The problem is nonlocal, quasilinear, and to leading order, is scaling invariant in the Sobolev space $H^{s_c}(\mathbb{R}^d)$ with $s_c=1+\frac d2$. We prove local well-posedness for large data in all subcritical Sobolev spaces $H^s(\mathbb{R}^d)$, $s>s_c$, allowing for initial interfaces whose curvatures are unbounded and, furthermore when $d=1$, not square integrable. To the best of our knowledge, this is the first large-data well-posedness result that covers all subcritical Sobolve spaces for the Muskat problem with surface tension. We reformulate the problem in terms of the Dirichlet-Neumann operator and use a paradifferential approach to reduce the problem to an explicit parabolic equation, which is of independent interest.